## DESCRIPTION
##   Vector Projections
## ENDDESCRIPTION

## Tagged by nhamblet

## DBsubject('Calculus')
## DBchapter('Vectors and the Geometry of Space')
## DBsection('The Dot Product')
## Date('6/3/2002')
## TitleText1('Calculus: Early Transcendentals')
## AuthorText1('Stewart')
## EditionText1('6')
## Section1('12.3')
## Problem1('41')
## TitleText2('Calculus: Early Transcendentals')
## AuthorText2('Rogawski')
## EditionText2('1')
## Section2('12.3')
## Problem2('41')
## KEYWORDS('Dot Product', 'Projection', 'Scalar', 'Vector', 'Orthogonal','vector')

DOCUMENT();

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGauxiliaryFunctions.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$a = non_zero_random(-10, 10);
$b = random(-10, 10);
$c = random(-10, 10);
$d = random(-10, 10);
$e = random(-10, 10);
$f = random(-10, 10);

$square = ($a)**2 + ($b)**2 + ($c)**2;
$adbecf = $a*$d + $b*$e + $c*$f;

$sclr = $adbecf / sqrt($square);

$vctr1 = $adbecf * $a / $square;
$vctr2 = $adbecf * $b / $square;
$vctr3 = $adbecf * $c / $square;

$orth1 = $d - $vctr1;
$orth2 = $e - $vctr2;
$orth3 = $f - $vctr3;

BEGIN_TEXT
$PAR
Let \( {\mathbf a} \) = ($a, $b, $c) and \( {\mathbf b} \) = ($d, $e, $f) be vectors. 
Find the scalar, vector, and orthogonal projections of \( {\mathbf b} \) onto \(
{\mathbf a} \).

$PAR
Scalar Projection: \{ ans_rule(30) \}
END_TEXT
ANS(num_cmp($sclr));

BEGIN_TEXT
$PAR Vector Projection: $BR (\{ ans_rule(30) \},
END_TEXT
ANS(num_cmp($vctr1));

BEGIN_TEXT
\{ ans_rule(30) \},
END_TEXT
ANS(num_cmp($vctr2));

BEGIN_TEXT
\{ ans_rule(30) \})
END_TEXT
ANS(num_cmp($vctr3));

BEGIN_TEXT
$PAR Orthogonal Projection: $BR (\{ ans_rule(30) \},
END_TEXT
ANS(num_cmp($orth1));

BEGIN_TEXT
\{ ans_rule(30) \},
END_TEXT
ANS(num_cmp($orth2));

BEGIN_TEXT
\{ ans_rule(30) \})
END_TEXT
ANS(num_cmp($orth3));

ENDDOCUMENT();